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Quiz 10: Partial derivatives and tangent planes

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Question 1

 
 
Which option correctly gives the two first order partial derivatives of the following function?
f(x,y) = ex + x-+ (2x+ y)4 y
a) x 1- 3 3 fx = e + y + 8(2x + y), fy = xlny + 4(2x + y)
b) f = ex + x2 + 8(2x + y)3, f = - x-+ 8(2x+ y)3 x y y y2
c) 2 fx = ex + x + 8(2x + y)3, fy = x lny + 8(2x + y)3 y
d) 1 x fx = ex +--+ 8(2x + y)3, fy = --2 + 4(2x+ y)3 y y
e) x2 x fx = ex +-- + 8(2x + y)3, fy = ex --2 + 8(2x+ y)3 y y

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Not correct. Choice (e) is false.
 

Question 2

 
 
Find the two first order partial derivatives, with respect to x and y, of
z = cos(x2y)+ sin y.
a) ∂z-= - 2xsin (x2y) ∂x and ∂z-= - x2ysin(x2y)- cosy ∂y
b) -∂z= - 2xysin(x2y) ∂x and ∂z= x2 sin(x2y)- cosy ∂y
c) ∂z ∂x-= 2x sin(x2y)- cosy and ∂z ∂y-= x2sin(x2y)
d) -∂z 2 ∂x = - 2xysin(x y) and ∂z- 2 2 ∂y = - x sin (x y)+ cosy
e) -∂z= - x2sin (x2y) ∂x and ∂z-= - 2xsin(x2y)+ cosy ∂y

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Not correct. Choice (e) is false.
 

Question 3

 
 
Find the first order partial derivative with respect to y of
2 f(x,y) = xexy + log(xy).
a) 2 2 1 1 exy + 2yx2exy + y-+ x-   b) 2 1 2yx2exy + y-
c) 2 1 xexy + xy-   d) 2 2 1 exy + y2xexy + x-
e) 2 xy2 1 2yx e + xy-

 

Not correct. Choice (a) is false.
Your answer is correct.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 4

 
 
Find the first order partial derivative with respect to x of
---1--- f(x,y) = x2 + y2.
a) 2x log(x2 + y2)   b) 2y - (x2 +-y2)2
c) 2x - (x2 +-y2)2   d) 2y log(x2 + y2)
e) 1 - (x2 +-y2)2

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 5

 
 
Find the tangent plane to the surface
2 2 z = x - y
at the point (5,-4,9).
a) z - 9 = 10(x - 5) + 8(y + 4)   b) z - 9 = 10(x - 5) - 8(y - 4)
c) z - 9 = 8(x + 4) + 10(y - 5)   d) z - 9 = 8(x- 5) - 10(y - 4)
e) z - 9 = 5(x- 8) + 4(y - 8)

 

Your answer is correct.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 6

 
 
The differential of the function
( ) -x2-- z = cos x+ y
is given by
a) 2 ( 2 ) dz-= - x-+-2xysin -x--- dx (x + y)2 x+ y .
b) 2 ( 2 ) 2 ( 2 ) dz = - x-+-2xy2 sin -x--- dx+ --x---2 cos--x-- dy (x + y) x+ y (x + y) x + y .
c) 2 ( 2 ) 2 ( 2 ) dz = - x-+-2xy2 sin -x--- dx+ --x---2 sin -x--- dy (x + y) x+ y (x + y) x + y .
d) x2 + 2xy ( x2 ) x2 ( x2 ) dz = --------2 sin ----- dy- -----2-sin ----- dx (x + y) x+ y (x + y) x + y .
e) ( x2 ) ( x2 ) dz = z - z0 = cos----- - cos ---0--- x+ y x0 + y0 .
f) None of the above.

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
Not correct. Choice (f) is false.
 

Question 7

 
 
If f(-1,3) = 4, fx(-1,3) = 5 and fy(-1,3) = -2, the linear approximation to f(-1.3,3.2) is given by:
a) 2.1   b) 4
c) 5.9   d) 4.1
e) there is not enough information given to estimate f(-1.3,3.2).

 

Your answer is correct.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 8

 
 
Using differentials, and without using a calculator, the approximate value of
2 3x-y f(x,y) = x e
at (2.25,5.5) is
a) 2   b) 4
c) 10   d) 18
e) 27

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Use f(x + dx,y + dy) = f(x,y) + fx(x,y)dx + fy(x,y)dy with x = 2, y = 6, dx = 0.25, dy = -0.5. In this way e3x-y = e0 and hence f(2,6), fx(2,6) and fy(2,6) can all be evaluated without a calculator. Actually response 4 is closer to the exact value but is not what is obtained when this method of approximation is used.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 9

 
 
Using differentials, the approximate volume of tin in a closed cylindrical tin can with radius 4cm, height 12cm and where the thickness of metal is 0.04cm, is
a) 3.84π cm3.   b) 5.12π cm3.
c) 24.8π cm3.   d) 16.226 cm3.
e) 12.8π cm3.

 

Not correct. Choice (a) is false.
Your answer is correct.
The volume of the tin is the difference between the outside volume and the inside volume of the tin which is approximately equal to the total differential of the inside volume. The volume of a cyclinder of height h and radius r is V = πr2h. Hence, the differential is
∂V- ∂V- 2 dV = ∂r dr + ∂h dh = 2πrh dr+ πr dh.
Now, dr = 0.04 and dh = 0.08 (Note that the cylinder is closed and so the dh = 0.08 comes from the contribution of the top and bottom of the cylinder which together are twice the thickness of the metal.) Therefore,
dV = 2π ⋅4⋅12 ⋅0.04+ π ⋅42 ⋅0.08 = 5.12π cm3.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 10

 
 
The volume of a circular cylinder is given by V = πr2h where r is the radius of the cylinder and h is its height. A circular cylinder with r = 3cm and h = 2cm has a volume of 18π cubic centimetres. The radius of the cylinder is now reduced by 0.5cm and the height by 0.2cm. An engineer makes an estimate of the reduction in volume using differentials. What is this estimate?
a) 6.75π cubic centimetres   b) 6.9π cubic centimetres
c) 7.8π cubic centimetres   d) 12.6π cubic centimetres
e) 21.21 cubic centimetres

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.